Zvalue

A zvalue, commonly written as z-value or z-score, is a standardized measure that indicates how many standard deviations an observed value is from the mean of a distribution. It is defined as z = (X − μ) / σ, where X is the value of interest, μ is the population mean, and σ is the population standard deviation. When μ and σ are unknown and estimated from data, the sample mean x̄ and sample standard deviation s are used, but the z-value interpretation assumes the underlying σ is known; otherwise a t-value is often used.

Interpretation and use: a z-value near zero shows the value is close to the mean; positive z-values are above the mean, negative values below. Under the normal distribution, z-values follow the standard normal distribution with mean 0 and variance 1, enabling direct calculation of p-values and confidence intervals. For a two-sided test, the p-value is 2 times the tail probability beyond |z|.

Applications: z-values are used to standardize data from different scales, compare observations from different distributions, perform z-tests, and construct confidence intervals when the population standard deviation is known. They also serve in outlier detection, with common heuristics such as |z| > 3 suggesting unusual observations under normality. In practice, data that are not roughly normal or that have unknown variance may require alternative methods, such as t-tests or nonparametric approaches.

Relation to related concepts: the z-value is a standard score or standardization of data, linked to the standard normal distribution, z-tests, p-values, and feature scaling in data analysis.